How to apply uv-mapping post-modifier?
Geometry generated from a modifier is not directly accessible until after the modifier is applied. Otherwise, we would be able to control each UV vertex of an object with a subsurf modifier before committing to it.

Please explain why you must keep the modifiers, so that we can help you find a change in workflow to answer your question. 

For now, the answer is that it is not possible to apply UV mapping to geometry made from a modifier that hasn't been applied. 

Here is an example of a possible work around that may meet the behavior of your 3ds Max reference. First, I'm making these assumptions about your workflow:


You are using a repeating texture that doesn't rely on precise UV
positioning.
You are creating assets for rendering purposes.
You are using Blender Render instead of Cycles (May work, I'm not
experienced with it.)


Instead of using a UV-map to determine the texture positioning , Assign a material and texture to the object and use its texture properties to map an image to generated coordinates using a cube projection.



The results will not be reflected in the view-port (You must use view-port shading set to material to see it), but creating a render will show the texture projected on all sides.